The last of the Fibonacci. George Havas Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra

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1 ProceedinRS of the Royal Society of Edinburgh, 83A, , 1979 The last of the Fibonacci groups George Havas Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra J. S. Richardson Department of Mathematics, University of Melbourne, Parkville, Victoria and Loon S. Sterling Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra (Communicated by Professor T. S. Blyth) (MS received 16 October Revised MS received 11 January Read 5 March 1979) SYNOPSIS All the Fibonacci groups in the family F(2, n) have been either fully identified or determined to be infinite, bar one, namely F(2,9). Using computer-aided techniques it is shown that F(2,9) has a quotient of order , and an explicit matrix representation for a quotient of order is given. This strongly suggests that F(2, 9) is infinite, but no proof of such a claim is available. 1. INTRODUCTION Conway [5] aroused interest in the Fibonacci groups in These groups have been studied in general by Johnson, Wamsley and Wright [10] and by Chalk and Johnson [4]. The Fibonacci group F(2, n) may be presented F(2, n) =(xb Xz,..., Xn; XIXZ = X3,.., Xn-ZXn-1 = Xn, Xn-1Xn = Xb XnX1 = xz). Determination of one of these groups was made as early as 1907 [11], and by 1974 [1] all bar one, namely F(2,9), had been either fully identified or determined to be infinite. Computer-aided techniques have been used in this investigation of F(2, 9). Computer implementations of group-theoretic algorithms utilized are a coset enumeration program [3], a Reidemeister-Schreier program [7], a nilpotent quotient algorithm program [12], an abelian decomposition program [9], and a Tietze transformation program. All the groups F(2, n) which are known to be finite have been determined and in fact can be identified by coset enumeration [see 8 for details of the most

2 200 Gearge Havas, J. S. Richardsan, and Lean S. Sterling difficult successful coset enumeration]. The group F(2, 9), which is known to have a maximal nilpotent quotient of order 152, resisted all attempts by coset enumeration. The reason for this is now clear. In this paper we show that F(2, 9) has a quotient of order We present the method for the discovery of this quotient, and also give an explicit matrix representation for a quotient of order The size of the largest quotient that we have been able to discover is governed by the availability of computer resources, and there is every reason to expect that much larger quotients exist. All indications suggest that F(2, 9) is infinite, but proof of such a claim eludes us. We thank L. G. Kovacs and M. F. Newman for many helpful discussions. We acknowledge the faithful performance of two Australian National University computers, a Univac 1100/42 and a DEC KA10, on which all machine calculations were done. 2. QUOTIENTS OF F(2, 9) AND ITS SUBGROUPS The abelian quotient of F(2,9) is isomorphic to Cz x Cz X C19, where Cn denotes a cyclic group of order n. The nilpotent quotient algorithm shows that F(2, 9) has a maximal nilpotent quotient isomorphic to Q x C19, of order 152 (here Q is the quaternion group). Using this information, it is easy to find presentations for subgroups of indices 2,4,8, 19, 38,76 and 152 in F(2,9). We hoped that the nature of the subgroups would cast light on F(2, 9). The simplest hope was that the abelian quotients of the subgroups would provide new information. Using a judicious combination of all the computer programs mentioned in 1 we were able to find the maximal abelian quotients of the subgroups of F(2, 9) corresponding to subgroups of Q x C19 In each case except the last, namely index 152, the maximal abelian quotient is the same as that of the corresponding subgroup of Q x C19 However the index 152 subgroup of F(2,9) has a maximal abelian quotient which is elementary abelian of order 518. The nilpotent quotient algorithm reveals that this subgroup has a maximal 5-quotient of class 3 with order 5741 This shows that F(2, 9) has a quotient of order Some of these computations are further described in [9]. The details of the computations outlined above are not particularly perspicuous. (They are available from the authors, as are programs for doing all the calculations.) However, it is possible to distil from the computer calculations a succinct demonstration of the existence of a quotient of F(2, 9) with order A MATRIX REPRESENTATION In this section we show that F(2, 9) = (Xl' Xz) has a quotient of order , which we exhibit as a linear group of dimension 19 over the field IFs of five elements.

3 The last of the Fibonacci groups 201 We begin by defining a number of matrices over 0=5. Let E GL9(5) 0 irreducible A19-1factors over of IF5); thelet polynomial so that Xl, X2 E GL1S(5); and for i = 1, 2 let where v1=(3,0,3,3,0,0, 3,2,1,0,0,0, 0,0,0, 0, 0, O)T v2=(1,0,0,0,0,0, 0,0,0,0,0,0, 0, 0, 0, 0, 0, O)T (- T denoting the transposed matrix). THEOREM. There is a homomorphism cf>: F(2, 9)~GL19(5) with image of order R, and such that cf>(xj = Yi (i = 1, 2). Proof. We note firstly that since B2 has order 19, the matrix group (B2) furnishes a 9-dimensional faithful irreducible representation over IF 5 of the cyclic group C19 If (Ui E GL2(5)), then (Uj, U2) furnishes a 2-dimensional faithful irreducible 0= 5-representation of the quaternion group Q. Since Xl and X2 are, respectively, Kronecker products of B2 with U1, and B10 with U2, it follows that G = (Xj, X2) furnishes an 18-dimensional representation of Q x C19' which is faithful because Q and C19 have coprime order, and irreducible since its tensor factors have coprime dimension [or, for example, by 6, Corollary 2.6, where we require the

4 202 George Havas, J. S. Richardson, and Leon S. Sterling fact that IFsis a splitting field for Q]. Furthermore, the epimorphisms F(2, 9)~C19=(B2), x1~b2, x2~b10 and give rise to a map F(2, 9)~Qx C19=G, Xi~Xi' which, it is easy to see, is again an epimorphism. Let V be an 18-dimensional vector space over IFs, on which GL1s(5) is considered to act from the left. In the light of the above, G acts irreducibly on V. The matrix group H = {[~ ~JE GL19(5): X E G, v E V} is isomorphic to the split extension of V by G, and has order IHI = IGI x IVI = We shall identify V with the (multiplicative) subgroup of H. We note that Yb Y2 EH. For i = 3, 4, 5,.., define Yi = Yi-21';-1' It may then be verified by direct calculation that In consequence there is a homomorphism To complete the proof it remains to be shown that (Yb Y2)=H. The action of G = (Xb X2) on V is realized in H via conjugation by the matrices y1l and y;:l. It follows that V n(y b Y2) is a submodule of the irreducible IFsG-module V (it should be remembered that "addition" 10 the module V is in fact matrix multiplication). A calculation shows that where [Y~, y2] 2 = y-2y-2y2y = [I l' YJ Y= (3, 0, 4,1,3,2,4,2,3,3,0,2,2,2,4,2,0, 2)T ~ O.

5 The last of the Fibonacci groups 203 Thus [Yi, 11] is a non-trivial element of Vn(y1, Yz) so Vn(Yb Yz)= V, that is, V::;(Yb Yz). Since (Yb Yz) has G as a quotient, we see that its order is divisible by Therefore (Y1, Yz) = H as required. 4. FURTHER CALCULA nons The group theory language Cayley [2] was used to investigate the matrix group H described above. Given the two generating matrices Y1 and Yz for H, Cayley confirmed that H has order During the computation permutation representations for H were calculated, including a lowest degree faithful representation which has degree 190. As a last try we looked at a corresponding index 190 subgroup of F(2, 9). This subgroup has a maximal abelian quotient isomorphic to C4, which is the same as for the corresponding subgroup of H. REFERENCES 1 A. M. Brunner The determination of Fibonacci groups. Bull. Austral. Math. Soc. 11 (1974), 2 J. J. Cannon. A draft description of the group theory language Cayley. SYMSAC ' R4 (Proc. ACM Sympos. Symbolic and Algebraic Computation, New York, Association for Computing Machinery, New York, 1976). 3 J. J. Cannon, L. A. Dimino, G. Havas and J. M. Watson. Implementation and analysis of the Todd-Coxeter Algorithm. Math. Comput. 27 (1973), C. P. Chalk and D. L. Johnson. The Fibonacci groups. 11. Proc. Ray. Soc. Edinburgh Sect. A 77 (1977), J. H. Conway. Advanced problem Amer. Math. Monthly 72 (1965), B. Fein. Representations of direct products of finite groups. Pacific 1. Math. 20 (1967), G. Havas. A Reidemeister-Schreier program. Proc. Second Internat. Conf. Theory of Groups, Australian Nat. Univ., Canberra, 1973, Lecture Notes in Mathematics 372 (Berlin: Springer, 1974). 8 G. Havas. Computer aided determination of a Fibonacci group. Bull. Austral. Math. Soc. 15 (1976) G. Havas and L. S. Sterling. Integer matrices and abelian groups. EUROSAM 79. Lecture Notes in Computer Science (Berlin: Springer), to appear. 10 D. L. Johnson, J. W. Wamsley and D. Wright. The Fibonacci groups. Proc. London Math. Soc. 29 (1974), G. A. Miller. The groups generated by three operators each of which is the product of the other two. Bull. Amer. Math. Sac. 13 (1907), M. F. Newman. Calculating presentations for certain kinds of quotient groups. SYMSAC '76, 2-8 (Proc. ACM Sympos. Symbolic and Algebraic Computation, New York, Association for Computing Machinery, New York, 1976). (Issued 21 September 1979)